Course for international guest/part time students

Faculty

Faculty of Science

Organization

TTK Department of Geometry

Code

alggeo1u0um17gm

Title

Algebraic geometry (p)

Usual semester

Spring

Published semester

2026/27/1

ECTS

3

Language

en

Learning outcomes

Knowledge: Knowledge of basic concepts, results and methods in the field. Ability: Application of knowledge in the field, understanding of interrelationships and problem solving. Attitude: Desire to improve mathematical knowledge and to learn as much as possible, and to apply knowledge as widely as possible. Autonomy and responsibility: Formulate and analyze mathematical questions independently and evaluate the limits of their applicability responsibly.

Course content

Introductory part: closed subsets of affine spaces and regular mappings, rational functions and mappings, quasiprojective varieties, finite mappings, dimension theory, product, irreducibility. Local theory: smooth and singular points, tangent space, local parameters, smooth subspaces, blow-ups and their properties, birational mappings, exceptional divisors, isomorphisms and birational equivalence, normal varieties and normalization. Divisors and differential forms: divisors, prime divisors, linear systems, divisor class group, line bundles, Picard group, differential forms, Riemann–Roch theorem for curves. Intersection numbers, applications in surface theory: intersection numbers, Bézout theorem, behavior in blow-ups, surface singularities. Possible advanced continuation: sheaves, homological algebra, sheaf cohomology, Spec(A), Zariski topology, abstract varieties. Possible topological addition: complex smooth projective manifolds, their homological properties, Lefschetz theorems and decomposition. Prerequisite knowledge: abstract algebraic structures (groups, rings, ideal theory, modules, fields, formal series), proficiency in calculating with them (localization, primary decomposition)

Assessment method

(written or oral) exam plus term mark

Bibliography

lecture notes

Recommended bibliography

R. Shafarevich: Basic Algebraic Geometry 1 and 2, Springer Verlag 1977, 1994 M.F. Atiyah, I.G. MacDonald: Commutative Algebra, Addison-Wesley Publ. Comp. 1969 M. Reid: Undergraduate Commutative Algebra, London Math. Soc. Students Texts 29, Cambridge Univ. Press 1995 J. Harris: Algebraic Geometry, A First Course, Graduate Texts in Math. 133, Springer-Verlag New York, 1992 R. Hartshorne: Algebraic Geometry, Graduate Texts in Math. 52, Springer-Verlag New York, 1977

Programmes of the course